Data driven approach for coupled electro-mechanical problems
Résumé
The classical approach in the computational mechanics considers the formulation of material modelswhose development rely on the data usually collected either through experimental measurements ornumerical simulation. The constitutive model should describe the data set as faithfully as possible,keeping at the same time it’s mathematical structure as simple as possible. Those two opposite demandsusually lead to material model that contains a number of simplifications and adjustments leading toempirical model which is less rich than the data used for it’s formulation. To overcome this informationloss, Kirchdoerfer and Ortiz [1] recently proposed a new approach which avoids the need of constitutivemodel in computational mechanics, replacing it by material data.In this work, we propose an application of the data driven approach (DDA) to the coupled, electrome-chanical behaviour. More precisely, we focus on the linear piezo-electric continuum whose governingequations are related to 1. mechanical (M) sub-problem described by elasticity, related kinematics andboundary conditions; 2. electrical (E) sub-problem related to the electrostatics, Gauss and Maxwell lawand related boundary conditions; 3. electro-mechanical coupling definedonly through the constitutive re-lations. As opposed to ’standard’ DDA related to elasticity [1] where the local state of the system in everymaterial point is characterized by set of stress-strain couples, here the material response is characterizedby 4-tuples composed of strain, stress and electrical field and electrical displacement. In DDA the solu-tion is characterized by electro-mechanical points that are ’the closest’ to material points, we propose anextension of the norm to metrize the distance. In addition, we give the details of the DDA algorithm forcoupled piezoelectric behaviour. Analogously as in standard DDA, the objective here is to findelectro-mechanical states(4-tuples) such that: (i) the strain and electrical fields are derived from displacementand electrical potential fields respecting the mechanical compatibility equations and Maxwell’s law and(ii) that the stress and electrical displacement verify the corresponding mechanical equilibrium as wellas Gauss law (dielectric equivalent). The coupling being only in terms of constitutive model, which weabandon, the great advantage of applying DDA approach to a class of coupled multiphysics problems isthat it results in twodecoupledsub-problems, the mechanical and dielectric problem. The performanceof the proposed extension is shown on few illustrative examples.
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